An n-dimensional vector space can be completely described by any set of "n" independent vectors. In general, the choice of a particular set of independent vectors is made for purposes of convenience. As an example, the same physical space may be described in terms of linear coordinates (x, y, z) for objects undergoing linear motion and radial coordinates (r, .THETA., z) for objects undergoing radial motion.
To transform coordinate descriptions between two vector spaces one may employ either transformation equations or empirical data tables, also known as "look-up tables" with the possible further step of interpolating between table entries. The method chosen depends on the circumstances. If simple transformation equations exist then these can be employed. However, often such equations are complicated, in which case the look-up table method should be considered. Look-up tables will be most easily applied to a finite vector space requiring only limited precision. As precision requirements increase, the table must either grow tremendously or else interpolation equations must be employed. There is a balance to be struck between the complexity of the interpolation equations and the exhaustiveness of the data table.
Vector spaces representing color are a specific example. It is now generally agreed that any color can be completely described by a three-dimensional vector space. That is to say, three independent colors can be combined to create all other colors. Two sets of colors are illustrative. For objects that are sources of light (such as color television sets) the traditional set of independent colors used is red, green and blue or RGB (the additive primary colors). For objects that absorb light, such as dyes, inks and paints, the traditional set of independent colors used is cyan, magenta and yellow or CMY (the subtractive primary colors).
Color printing is normally carried out using three or more ink colors. Usually four printing inks known as the "process colors" (cyan, magenta, yellow and black or CMYK) are used. While inks combine in a predictable, repeatable way, the relationships are nonlinear. Therefore, black ink is frequently used directly (instead of indirectly as a combination of the CMY inks) because black is usually an important color that must be precisely controlled.
Sometimes, particularly in the packaging industry, "non-process" inks are used to directly control specific colors, such as brown or orange, that may be important in a given printing. As with the use of black ink, these non-CMY inks are used by the printer to keep better control of the printing process and to obtain a better color match on the final printed result.
FIG. 1 illustrates-a typical color printing installation in block diagram form. A color transparency 10 is entered into the color printing system using an input device, such as a scanner 14. A scanned image is manipulated using a page composition system 16. Intermediate results can be viewed on such proofing devices 18 as a monitor 20 (at low resolution), a digital film recorder 21 (at high resolution) or an ink jet printer 22. The final product can be printed on a color printing device 24.
FIG. 1 also illustrates a major complication in the printing process, i.e. color space transformations. The fact that different devices in a color printing installation work with different color space descriptions necessitates the use of color converters 26, 28, 29 and 30 between devices.
Although color space is three dimensional, FIG. 1 illustrates transformations between larger coordinate systems. The transformation at 26 is from three space to five space. The transformations 28 and 29 are from five space to three space. The transformation at 30 is from five space to four space. These more complicated transformations necessitated by the physical properties of printing inks involve mathematically superfluous dependent basis vectors as well as the three independent basis vectors required to describe color space.
There are three cases to consider. First, the case where a color space defined in terms of only independent basis vectors is transformed into a color space defined in terms of both independent and dependent basis vectors. Second, the case where a color space defined in terms of both independent and dependent basis vectors is transformed into a color space defined in terms of only independent basis vectors. Third, the general case where a color space defined in terms of both independent and dependent basis vectors is transformed into a color space defined in terms of both independent and dependent basis vectors.
In the first case, the conventional transformation method is to input the original vector into a look-up table to determine the image vector in terms of only the independent basis vectors. These independent components are then further processed using color extraction optimising algorithms, such as those described in U.S. Pat. No. 4,879,594, to produce the dependent components and as a result to modify the independent components. As these algorithms are time consuming, real time processing is sacrificed using this method.
In the second case, the conventional transformation method is to use a look-up table with as many dimensions as there are basis vectors, dependent and independent, in the original vector space. This table can then directly supply the independent components of an image vector. This method tends to be impractical because look-up tables increase in size exponentially with each new input parameter and processing time for interpolation equations also increases dramatically with each new input parameter.
The third case is merely a combination of the first two cases. The same problems are applicable.